Games Theory and Multinomial Real Options Model to assess strategies, agreements and penalties


  • Gastón Silverio Milanesi Departamento Ciencias de la Administración, Universidad Nacional del Sur, Argentina


When selecting and designing strategies in competitive environments, three potential sources of uncertainty must be taken into account: risks deriving from self-actions, risks emerging from states of nature and risks present in competitors’ decisions. That is why a numerical model that considers one’s competitors’ actions is needed to develop value strategies, joint venture design and penalty analysis. This paper proposes a simple numerical Game Theory and Real Options model with multiple sources of underlying risk. The first part sets forth the mathematical basis of the model, illustrating its function with case valuations related to strategies where there is no collaborative agreement. Cooperation strategies and default monetary penalties are then valued, and finally the main conclusions are drawn.


Strategy assessment, Multinomial real options, Game theory, Initiative strategy, Strategic agreement, Penalty assessment


Aguado-Franco, J. C. (2007). Teoría de la decisión y de los juegos. Madrid: Delta Publicaciones.

Armada, M., Kryzanowski, L & Pereira, P. (2009). Optimal investment decisions for two positioned firms competing in a duopoly market with hidden competitors. European Financial Management, 17, 305-330.

Axelrod, R. (1981). The Emergence of Cooperation among Egoists. The American Political Science Review, 75(2), 306-318.

Axelrod, R. (1986). La Evolución de la Cooperación. Madrid, España: Alianza Editoria S.A.

Black, F. & Scholes, M. (1972). The Valuation of Options Contracts and a Test of Market Efficiency. Journal of Finance, 399-418.

Boyer, M., Laserrere, P. & Moreaux, M. (2012). A dynamic duopoy investement game without commitment under uncertainty market expansion. Interntational Journal of Industrial Organization, 30, 663-681.

Boyle, P. (1988). A lattice framework for option pricing with two state variables. Journal of Finance and Quantitative Analysis, 23, 1-12.

Brandao, L. & Dyer, J. (2009). Projetos de Opcoes Reis com Incertezas Correlacionadas. Revista de Administracao e Contabilidade da Unisinos(1), 19-26.

Brandao, L., Dyer, J. & Hahnn, W. (2012). Volatility estimation for stochastic project value models. European Journal of Operational Research, 220(3), 642-648.

Brous, P. (2011). Valuing an Early-Stage Biotechnology Investment as a Rainbow Option. Journal of Applied Corporate Finance, 23(2), 94-103.

Castro-Monge, E. (2010). El estudio de casos como metodología de investigación y su importancia en la dirección y administración de empresas. Revista Nacional de Administración, 2(1), 31-54.

Chance, D. (2007). A Synthesis of Binomial Option Pricing Models for Lognormally Distributed Assets. SSRN, 1-25.

Chance, D. (2008). A Synthesis of Binomial Option Pricing Models for Lognormally Distributed Assets. Journal of Applied Finance, 18(1), 38-56.

Copeland, T & Antikarov, V. (2003). Real Options: a practitioner ́s guide. New York: Cengage Learning.

Cox, J., Ross, S & Rubinstein, M. (1979). Option Pricing: A Simplified Approach. Journal of Financial Economics, 7, 229-263.Culik, M. (2016). Real options valuation with changing volatility. Pespectives in Science, 7, 10-18.

Derman, E., Kani, I & Chriss, N. (1996). Implied Trinomial Trees of the Volatility Smile. (Goldman-Sachs, Ed.) Quantitative strategies research notes.

Dixit, A. & Pindyck, R. (1994). Investment under Uncertainty (1 ed.). New Jersey: Pricenton University Press.

Dixit, A. & Nalebuff, B. (1991). Thinking Strategically: The competitive edge in business, politics and everyday life. New York, EE.UU: Norton Press.

Fudenberg D. & Tirole, J. (1985). Preemption and rent equalization in the adoption of new technology. Review of Economics Studies, 52(3), 383-401.

Fudenberg, D. & Tirole, J. (1986). A theory of exit in doupoly. Econometrica, 54(4), 943-960.

Gamba, A. & Trigeorgis, L. (2007). An Improved Binomial Lattice Method for Multi-Dimensional Options. Applied Mathematical Finance, 14(5), 453-475.

Ghemawat, P. & Nalebuff, B. (1985). Exit. Journals of Economics, 16(2), 184-194.

Graham, J. (2011). Strategic real options under asymmetric information. Journal of Economics and Dynamic Control, 35(6), 922-934.

Grenadier, S. (1996). The strategic exercise of options: Development cascades and overbuilding in real estate markets. Journal of Finance, 51(5), 1653-1679.

Grenadier, S. (2000). Options exercise games: The intersection of real options and game theory. Journal of Applied Corporate Finance, 13(2), 99-107.

Grenadier, S. (2002). Option exercise games: an application to the equilibrium investement strategies of firms. Review of Financial Studies, 15(3), 691-721.

Guintis, H. (2009). Game Theory Evolving (2 ed.). United Kingdom: Princeton University Press.

Haahtela, T. (2011) (a). Displaced Diffusion Binomial Tree for Real Option Valuation. SSRN, 1-30.

Haahtela, T. (2011) (b). Recombining trinomial tree for real option valuation with changing volatility. Annual Real Options Conference., 1-25.

Hsu, Y. & Lambrecht, B. (2007). Pre-emptive patenting under uncertatinty and asymmetric information. Annals of Operations Research, 151, 5-28.

Korn, R. & Muller, S. (2009). The decoupling approach to binomial pricing of multi-asset options. The Journal of Computational Finance, 12(3), 1-30.

Kreps, D. (1982). Rational Cooperation in Finitely Repeated Prisioners ́ Dilemmas. Journal of Economics Theory, 27, 245-252.

Kulatilaka, N. & Perotti, E. (1998). Strategic growth options. Management Science, 44(8), 1021-1031.Lambrecht, B. (2001). The impact of debt financing on entry and exit in duolpoly. Review of Financial Studies, 14(3), 765-804.

Lambrecht, B. & Perraudin, W. (2003). Real options and preemption under incomplete information. Journal of Economics Dynamics and Control, 27, 619-643.

Lari-Lavassani, A., Simchi, M. & Ware, A. (2001). A discrete valuation of swing options. Canadian applied mathematics quaterly, 9(1), 35-73.

Medina-Tamayo, R & Rodriguez-Pinzon, Y. (2010). Una revisión de los modelos de volatilidad estocástica. Comunicaciones en Estadística, 3(1), 79-97.

Merton, R. (1973). The Theory of Rational Options Princing. Bell Journal of Economics and Management Science, 141-183.

Milanesi, G. (2013). Asimetría y Curtosis en el Modelo Binomial para valora Opciones Reales: caso de aplicación para empresas de base tecnológica. Estudios Gerenciales Journal of Management and Economics for Iberoamerica, 29(128), 368-378.

Milanesi, G. (2021). Opciones reales secuenciales cuadrinomiales y volatilidad cambiante: Incertidumbres tecnológicas y de mercado en desarrollos de inversiones biotecnológicas. Revista Méxicana de Economía y Finanzas (REMEF), 24-49.

Milanesi, G., Pesce, G., & El Alabi, E. (2013). Technology-Based Start up Valuation using Real Opciones with Edgeworth Expansion. Journal of Financial and Accounting,1(2), 54-61.

Milanesi, G., Pesce, G., & El Alabi, E. (2014). Valoración de empresas de base tecnológica: Análisis de riesgo y el modelo binomial desplazado. Revista Española de Capital de Riesgo(4), 15-24.

Milanesi, G., & Tohmé F. (2014). Árboles Binomiales Implícitos, Momentos Estocásticos de Orden Superior y Valuación de Opciones. Revista de Economía Política (REPBA), 12(7), 45-72.

Milanesi, G., & Thomé F. (2015). Un modelo consolidado de opciones reales, teoría de juegos y análisis de costos de transacción para el diseño de acuerdos contractuales. Revista de Economía Política de Buenos Aires, 14, 59-81.

Murto, P. (2004). Exit in duopoly under uncertainty. Journal of Economics, 35(1), 111-127.

Nash, J. (1953). Two-Person Cooperative Games. Econometrica, 21(1), 128-140.

Num, J. (2015). Real Options Analysis (Third Edition): Tools and Techniques for Valuing Strategic Investments and Decisions with Integrated Risk Management and Advanced Quantitative Decision Analytics (3 ed.). CreateSpace Independent Publishing Platform.

Pareja-Vasseur, J, Prada-Sánchez, M & Moreno-Escobar, M. (2019). Volatilidad en Opciones Reales: Revisión literaria y un caso de aplicación al sector petrolero colombiano. Revista de Métodos Cuantitativos para la Economía y la Empresa(27), 136-155.

Pawlina, G., & Kort, P. (2006). Real options in an asymmetric duopoly: Who benefits from your competitive disadvantage? Journal of Economics and Management Strategy, 15(1), 1-35.

Paxson, D., & Melmane, A. (2009). Multi factor competitive internet strategy evaluations: Search expansion, portal synergies. Journal of Modeling Management, 4(3), 249-273.

Paxson, D., & Pinto, H. (2003). Rivalry under price and quantity uncertainty. Review of Financial Economics, 14, 209-224.

Rendleman, R., & Bartter, B. (1979). Two State Option Pricing. The Journal of Finance, 34(5), 1093-1110.

Rubinstein, M. (1983). Displaced Diffusion Option Pricing. Journal of Finance, 38(1), 213-217.

Rubinstein, M. (1994). Implied Binomial Trees. Journal of Finance, 49, 771-818.

Rubinstein, M. (1998). Edgeworth Binomial Trees. Journal of Derivatives (5), 20-27.

Rubinstein, M. (2000). On the Relation Between Binomial and Trinomial Option Pricing Model. Berkeley, Research Program in Finance-292. California: UC Berkeley.

Smit, H. (2003). Infraestructure investment as a real options game: The case of European airport expansion. Financial Management, Winter, 5-35.

Smit, H., & Ankum, L. (1993). A real options and game-theoretic approach to corporate investment strategy under competition. Financial Management, 22(3), 241-250.

Smit, H. & Trigeorgis, L. (2004). Strategic Investment: Real Options and Games (1 ed.). New Jersey, Estados Unidos: Princeton University Press.

Smith, J. (2005). Alternative Approach for Solving Real Options Problems. Decision Analysis, (2), 89-102.

Smith, J., & Nau, R. (1995). Valuing Risky Projects: Option Pricing Theory and Decision Anaysis. Management Science (5), 795-816.

Thijssen, J. (2010). Preemption in a real option game with a first mover advantage and a player-specific uncertainty. Journal of Economics Theory, 145, 2448-2462.

Trigeorgis, L. (1995). Real Options in Capital Investment: Models, Strategies and Applications (1 ed.). London, United Kindgon: Praeger.

Van der Hoek, & Elliot, R. (2006). Binomial models in Finance. New York, United State: Springer Science.

Wilmott, P. (2009). Frequently Asked Questions in Quantitative Finance (Segunda ed.). United Kingdom: John Wiley & Sons.

Zapata-Quimbayo, C. (2019). Valoración de opciones reales con múltiples incertidumbres mediante modelos K dimensionales. ODEON, 16, 97-121.